# Number density

The number density is a quantity that describes how many countable things of a type occur per volume element . These things can be objects, but also abstract entities such as events . As with all densities , the number density is an intensive quantity .

Most often the number density of objects in a volume of space is of interest. Therefore the definition is given below for this quantity. Based on this model, it is easy to create them in the same way for flat or curved surfaces or one-dimensional structures (lines, polygons, flat curves, spatial curves).

## Overview through examples

In fields such as physics , astronomy , chemistry , biology , ecology , particles in the narrower sense such as atoms, molecules, electrons, nucleons, photons or stars, galaxies, etc. are meant. The number density is then given as the number of these particles contained in an area of ​​physical space divided by the volume or the surface area or the length of the area under consideration. These cases can, if necessary, by the name of spatial number density ( english volume number density ) area-based number density ( english Area number density ) or length-related number density ( English Line number density are) distinguished.

The observed spatial area can be defined by the objects themselves, such as the boundaries of a solid , or simply any part of physical space. The spatial area can be open, so it does not necessarily have to have a material boundary, a closed material surface.

In areas such as population geography , demography , social sciences , individuals or subjects are counted. With population density , the area-based number density of the population of a state, a city is called so. The individuals are the inhabitants , their area-related density is the population density .

### Atomic number densities of the chemical elements

Number densities of the atoms of the chemical elements at 20 ° C and 1000 hPa. The number densities of the atoms of the 11 gaseous elements at this temperature and pressure are of the order of 10 19 atoms per cubic centimeter and therefore appear in the figure as if they were zero.

The figure shows the number densities of the atoms (atomic number densities) of the chemical elements at 20 ° C as an example of the physical quantity number density . It should be emphasized that carbon occurs naturally in pure form as diamond and graphite . These carbon modifications have significantly different mass densities and thus also atom number densities . Their atomic number densities are highlighted in the figure as a blue or purple point. Carbon in the diamond modification has by far the greatest atomic number density of all elements.

### Atomic number density of an ideal gas

In the case of a gas, in addition to the temperature, the pressure of the gas determines the atomic number density, with a standard condition according to the dimensional reference temperature for physical quantities being selected here as an example . Let the temperature be 20 ° C, the absolute temperature consequently the pressure . For an ideal gas , the atomic number density is about ${\ displaystyle T = 293 {,} 15 \ \ mathrm {K}}$ ${\ displaystyle p = 1.01325 \ cdot 10 ^ {5} \ \ mathrm {Pa}}$${\ displaystyle n}$

${\ displaystyle n = {\ frac {p} {k _ {\ mathrm {B}} \ cdot T}}}$.

It is the Boltzmann constant . If the numerical values ​​are used in this formula, the atomic number density of the ideal gas results in ${\ displaystyle k _ {\ mathrm {B}} = 1 {,} 38065 \ cdot 10 ^ {- 23} \ \ mathrm {J / K}}$

${\ displaystyle n = 2 {,} 5035 \ cdot 10 ^ {19} {\ frac {\ text {Atoms}} {\ mathrm {cm ^ {3}}}}}$

(see also Avogadro's theorem ). The atomic number densities given above for the 6 naturally occurring noble gases deviate only slightly from the atomic number density of the ideal gas. The size difference is in helium with note to. ${\ displaystyle n = 2 {,} 5577 \ cdot 10 ^ {19} {\ text {Atoms}} / \ mathrm {cm ^ {3}}}$

## definition

The spatial number density is equal to the number of objects contained in a spatial area divided by the volume of the spatial area ${\ displaystyle n}$${\ displaystyle \ mathrm {d} N}$${\ displaystyle \ mathrm {d} V}$

${\ displaystyle n ({\ vec {x}}) = {\ frac {\ mathrm {d} N} {\ mathrm {d} V}}}$

Since this mathematically strict definition of the number density is generally impractical - for atoms assumed to be point-shaped in a crystal, the number density would manifest itself as a Dirac comb , for example - the number density in areas in which macroscopic quantities are examined does not have an infinitesimal volume element , but rather defined over a mesoscopic spatial area . This mesoscopic spatial area must be selected in such a way that an averaging of the quantities in this area has no influence on the macroscopic physics. ${\ displaystyle \ Delta V}$

The arithmetic mean of a spatially varying number density in the spatial area with the volume is given by ${\ displaystyle {\ overline {n}}}$${\ displaystyle V}$

${\ displaystyle {\ overline {n}} = {\ frac {\ iiint _ {V} n ({\ vec {x}}) \, \ mathrm {d} V} {\ iiint _ {V} \ mathrm { d} V}} = {\ frac {N} {V}}}$

certainly. The number of all objects in a space with the volume is calculated by integration, ${\ displaystyle N}$${\ displaystyle V}$

${\ displaystyle N = \ iiint _ {V} n ({\ vec {x}}) \, \ mathrm {d} V,}$,

where the volume element is the space region. If all objects have the same mass , the total mass of all objects in the space is with the volume${\ displaystyle \ mathrm {d} V}$${\ displaystyle m_ {0}}$${\ displaystyle M}$${\ displaystyle V}$

${\ displaystyle M = m_ {0} \ iiint _ {V} n ({\ vec {x}}) \, \ mathrm {d} V.}$.

Similar formulas apply to other extensive quantities that are formed from number densities of countable objects. For example, for electrical charge , mass is replaced by the total charge and the mass of an object is replaced by the charge of an object in the last formula. ${\ displaystyle M}$${\ displaystyle Q}$${\ displaystyle m_ {0}}$${\ displaystyle q_ {0}}$

### unit

The SI unit of spatial number density is m −3 , but the unit cm −3 is often used.

Included in the numerator physical quantity number is provided as a size of the number of dimensions in the SI unit system, no unit associated with the addition of a auxiliary word , such as "play", "unit [s]", "pair", "set" or the name of the counted Objects / subjects (such as 12 trees or 24 particles ) are however tolerated.

## Examples of name variants

There are some variants of the name that mean a size number density . Two examples should be mentioned here.

### Object density

The following three names variants are found in the literature: number density of objects , object number density , object density . If the objects particles, this leads Name Alternate number density of the particles , particle number density , particle , for example, the number density of the atoms , atomic number density , atomic density . Only in the first variant of each name does the plural already express in the name that it is a quantity for many objects, particles or atoms. The third name variants are ambiguous. It could also mean the mass density of an object, particle or atom. There are synonyms for the object, the name variants proliferate, for example, if instead of particles of particles talks.

## Examples of how number densities are determined

There are different methods of determining the number density. If number densities of several types of objects / subjects are to be determined, this can be restricted to one type in most cases . In the following it is assumed that only one type is contained in the space area.

### Determine number density by counting

Regardless of which type of number density is to be calculated, it starts with counting . Even when counting with the index finger, people use their ability to abstract .

A number density of ecology is the population density or stand density. It considers the size of a population in a spatial area or an area. Examples of the objects to be counted are fish or plankton organisms in a lake, bacteria in a tank or soil organisms. Area-related population densities are given in units of individuals per square meter, per hectare , etc. Bird population densities are particularly determined in the vicinity of airports. In the case of the population density of a species especially in its settlement area, the number density is called abundance .

### Determine number density by mass comparison

Each of the objects in the spatial area has the mass and the total mass of all objects . In order to count the objects in the spatial area, the mass of all objects and the mass of a single object can be measured in classical mechanics and the number of objects can be calculated from this: ${\ displaystyle m_ {0}}$${\ displaystyle M}$${\ displaystyle N}$

${\ displaystyle N = {\ frac {M} {m_ {0}}}}$

Once the volume of the spatial area has been determined, the number density follows${\ displaystyle V}$${\ displaystyle n}$

${\ displaystyle n = {\ frac {M} {V \ cdot m_ {0}}}}$.

A common application of this formula is the calculation of the number density for atoms with a given mass density and the mass of an atom in a given space.

## Individual evidence

1. Units and terms for physical quantities: standards . 9th edition, as of the repr. Standards: January 2009. Beuth, Berlin 2009, ISBN 978-3-410-17239-0 (684 pages).
2. Formula symbols, formula set, mathematical symbols and terms: standards . 3rd edition, status of the printed standards: January 2009. Beuth, Berlin, Vienna, Zurich 2009, ISBN 978-3-410-17244-4 .
3. DIN 25401: Terms of nuclear technology, only on CD-ROM . Beuth, December 2015 ( online ). In the last printed edition of the standard, the previous edition of 2002-04, is on page 19 under item 3103 in addition to the current name of the size of neutron number density also the former name neutron density specified.
4. Lexicon of Physics - Spectrum of Science. Retrieved February 12, 2018 .